Motivation

A number of challenges in atomic structure in the coming decades
lie where the techniques of the past appear to be wanting. The most
notable ‘up to the minute’ exigencies that have taken center stage are the
complications of microscopic specimen size and soft order [1,2]. The
last problem surfaced with Dan Shechtman’s discovery of quasi-crystals
[3,4]; systems in which the diffraction pattern defy Bravais classification
or the assignment of Miller indices. It exposed a severe inadequacy in
Laue-Bragg’s famously successful technique in crystallography [5-9].
Furthermore new advances such as high intensity coherent radiation
beams, ultrashort exposure and extremely fast detectors, coupled with
the desire for atomic resolution in real time, demands a reassessment of
the premises of classical crystallography; also it is imperative to find the
way to an alternate approach that is indifferent to long-range order and
baggage from the associated reciprocal space, leverages the principle
of ‘reflection-diffraction duality’ to embody Bragg’s computationally
facile top-down approach with Laue’s local bottom-up perspective, and
(ii) covers a definition of crystal matter that is more inclusive especially
the complex regime of partial loosening of order, but not complete
disorder.

Overview

The Nobel winning discovery of ‘Laue spots’ in 1912 is one of the
pivotal breakthroughs in history [10]; to this day, over a century later
its implications continue to reverberate through the sciences. In 1925
Alver Gullstrand, Chairman of the Royal Swedish Academy of Sciences
would declare [11], “ This epoch-making discovery [Laue spots], which
not only bore upon the nature of X-radiation and the reality of the
space lattice assumed in crystallography, but also placed a new means
of research into the hands of Science …”. Paradoxically, although Max
Laue anticipated the ‘Laue effect’, but his new 3-dimensional vector
diffraction theory had some serious flaws [12]. As a matter of fact, this
theory shockingly misinterpreted the observed patterns. Nevertheless
out of Max Laue’s fecund but imperfect analogy of three-dimensional
diffraction gratings, wave scattering has emerged as a blockbuster
success in a wide range of technologies including low energy
electron diffraction (LEED), x-ray crystallography (XRD), and x-ray
fluorescence (XRF). Techniques helped reveal the geometric structures
of the simplest salts as well as the complex double helix spirals of the
two strands of polynucleotides in life’s genetic code, deoxyribonucleic
acid (DNA), netting dozens of Nobel Prizes in Physics, Chemistry and
medicine.

A surprise came in the early 1980’s with Dan Shechtman’s
perplexing discovery [3,4] of spot patterns that disturbed the certitude
of classical crystallography. This quasi-crystal puzzle would force the
International Union the International Union of Crystallography to a
redefinition of crystals in 1991 [13]. And in 2011 Shechtman would
be honored with the Nobel Prize for chemistry. Here we query where
Laue’s mathematics slipped and where Bragg’s formula really and
exactly fits into classical crystallography. Especially re-examine the
legacy of Miller indices and the dual space and prompt an improved
approach that accommodates current trends and future needs of structure analysis; particularly the concerns of imperfect order, realspace
configuration, microscopic specimen size, and extremely short
exposures to high intensity beams [1,2,14].

Introduction

In 1912 Max Laue an expert and lecturer of optics in Arnold
Sommerfeld’s theoretical physics Institute at Ludwig-Maximilians-
University (LMU) in Munich, Germany, theorized that if solid crystals
comprise of ordered, three-dimensional packing of atoms, then
diffraction spots would be produced by passing x-ray waves through crystals.. In his Nobel Prize lecture [10] Max von Laue
described the run up as

“the acknowledged masters of our science, … entertained certain
doubts about this viewpoint…. A certain amount of diplomacy was
necessary before Friedrich and Knipping were finally permitted to carry
out the experiment “.

After Laue’s presentation at the Berlin Physical Society on June
8 1912, his former professor Max Planck, a close scientific associate
and long-time friend Albert Einstein, the general relativist Karl
Schwarzschild (of Black hole fame) and others applauded Laue’s
prescience.

Albert Einstein

To explain his effect, Max Laue an authority in physical optics proposed that real crystals coherently scatter off x-ray waves, as would
a three-dimensional diffraction grating. Notice, that until this time only
one and two-dimensional cross gratings were in existence, although
the later was of little practical use. But a 3-d grating was totally a new
invention. In his Nobel lecture quoted earlier, [10] Laue describes it as

“at that point that my intuition for optics suddenly gave me the
answer: lattice spectra would have to ensue”.

Guided by an analogy with one and two-dimensional optical
gratings ‘generalized’ to the third dimension, (Figure 1) Laue demanded
that at each spot the wave and lattice vectors must satisfy the following
set of three ‘fundamental equations’, as follows

Figure 1: Laue diffraction kernel in an orthorhombic crystal and a ray (OO”)
in three-dimensional Euclidean space. The four atoms, the Cartesian axes
and the ray define three planes, namely XAOO”, YBOO” and ZCOO”,
the angular coordinates of OO” are the three angles (α, β, γ). The lattice
vectors are a, b and c respectively.

(1)

Likewise

(2)

and

(3)

The ‘dot’ in eqs. 1-3, stand for scalar product, angles are measured
with respect to the crystal axes and the Laue indices or order numbers
nx, ny and nz, are three independent (typically small but not necessarily
distinct) integers. In Laue’s bottomup description, short-range local
order is paramount; the effect of the entire crystal is modeled by the actions of just the few atoms included in the
kernel.

Laue’s Achilles heel

Incredibly, it was Max Laue’s own 3-d diffraction grating theory
that was problematic and consequently failed to interpret Friedrich and
Knipping’s experimental data [12]. As a matter of fact simultaneously
satisfying the dispersive equations along all the three Cartesian axes
was overwhelming and Laue’s results came out erroneous, including a
wrong answer for the crystal density! Incidentally, Laue was aware of
some of the problems with his theory, because in his Nobel lecture he
stated,

” … the three specified numbers… I made no secret of the fact that I
could not attribute to these values the same degree of reliability…”

But Max Laue did not realize that he did not properly constrained
the system of equations and lacked Euclidean length invariance; so the
theory was producing too many solutions [12]. As a matter of fact, it is
not generally known that Euclidean geometry and dimensionality of the
space-time have a significant influence on what physical phenomena
are permissible; vector product and Huygen’s principle for wave are
two classic peculiarities of 3+1 dimensional space-time [15]. In 3+1
dimensional space-time, generalizing from one-dimensional to
two-dimensional gratings is ‘trivial’ because the angular position of
any maximum is completely determined by the grating constant in
that particular direction. For example, two cross gratings with same
grating constant, b, along y direction but different values of a (along
the x-axis), the nx=2 and ny=1 maximum will be produced, at exactly
the same angle β, but at two different values of α, as determined by
equation 1. The reader will immediately notice that the angle γ wrt
to the z-axis is also different because the angles α, β and γ are not
independent, but interlinked by the square of the direction cosine
rule, as follows,

(4)

In essence in the space we live in it is impossible to impose three
Cartesian components independently of each other as seemingly
required by Laue’s three fundamental equations 1-3. Consequently,
there is a critical theoretical difference between 2-d and 3-d gratings.
Allvar Gullstrand, [11] quipped

“Inasmuch as this is a three-dimensional grating, its effect is in
essential respects unlike the effect of the previously known line and cross
gratings”.

Reaction to Laue’s Effect

William Henry Bragg (WHB) the erstwhile professor of physics at
Adelaide, Australia and an acknowledge authority in x-rays, who by
this time was a physics professor at Leeds, in UK along with his son
William Lawrence Bragg (WLB) took notice of Laue’s effect [16-20]. So
did Ernest Rutherford’s atomic group at Manchester [21,22]. Especially
Rutherford’s protégé the young and brilliant Henry Gwyn Jeffreys
Moseley, Henry Moseley was particularly critical of Laue’s theory and
in a letter to his mother, he wrote “[Laue] gave an explanation which
was obviously wrong”.

Truth be told, Moseley was notorious for his critical attitude
towards fellow researchers as-well-as his own doctoral advisor
Earnest Rutherford, ‘son of a flax farmer’ from New Zealand, a small
inconsequential colony in the remotest frontier of the then vast
British Empire. Nevertheless Moseley’s self-confidence and genius
was recognized at Manchester and despite his initial hesitations about
jumping on the crystal structure band wagon, Rutherford would let
Moseley temporally move to Leeds to learn x-ray spectroscopy, then
being invented and developed by WHB. Perhaps not so coincidentally,
at the very same time June-August of 1912 Rutherford’s visiting ‘postdoc’
Niles Bohr, would also abruptly change his previous research
plan and get busy with a new theory for a stable quantum atom.
Mosley made excellent use of his stay at Leeds and upon his return to
Rutherford’s laboratory back in Manchester atom he would soon invent
x-ray fluorescent spectroscopy (XFS) and apply it to discover his name
sake Moseley’s Law, which relates the frequency (energy) of Barkla’s
characteristic x-ray emission from the atom with the nuclear charge
of the chemical element. As a matter of fact in the process discovering
atomic number (Z) and identifying Z with the nuclear charge of the
atom. He was a nominee for the Physics Nobel Prize in both 1914 and
1915 and certainly would have been a recipient; because in the same
(1925) lecture [11] Alvar Gulstrand also announced that

“the greatest success by the young scientist Moseley… He further
discovered … what is known as the atomic number, … has proved to
distinguish the elements better than the atomic weight … Moseley fell at
the Dardanelles before he could be awarded the prize “.

Henry Moseley is recognized as one of the youngest pioneers in
nuclear physics. However, it was WLB who proffered a new explanation
for Laue’s effect that correctly accounted for all the observations and
paved the way to atomic crystallography. Incidentally, at that time
Bragg was also a student of Joseph John Thomson (Cavendish) and
William Jackson Pope (Chemistry) at Cambridge University. Earlier
Professor Pope [23] along with the well-known British amateur
geologists William Barlow [24] had already developed close packed
model structures of various crystals. Barlow was also the first to
note the differences between simple cubic and of face centered cubic
arrangements of atoms, a point that Laue had completely missed in his
analysis, described earlier (Figure 2). William Lawrence Bragg had an empirical and more hands on approach. Noting that when he turned
the crystal target by 3o, the whole spot pattern rigidly rotated by 6o, and
knowing that generally diffraction patterns follow some complicated
trigonometric dependence, Bragg realized that certain planner surfaces
in the crystal must be (partially) reflecting off the x-rays. Apparently, C.
T. R. Wilson of cloud chamber fame had a part in this, because in his
first paper [16] published in November of 1912 WLB writes,

“… it was suggested to me by Mr. C.T.R. Wilson that crystals with
very distinct cleavage planes, such as mica, might possibly show strong
specular reflection of the rays. On trying the experiment it was found
that this was so… left no doubt that the laws of reflection were obeyed…
bending the mica into an arc, the reflected rays can be brought to a line
focus … yet the effect almost certainly not a surface one… ”.

Bragg’s focus was on the large-scale geometry, specifically atomic
planes but not on local atomic order. He introduced the practice
of identifying a spot by the Miller indices (h, k, l); he also proffered
the eponymous formula or Bragg’s law for the wavelength of x-ray
associated with a spot as,

(5)

Where θ is the ‘glancing angle’ and dB the inter-planer distance
and n is the Bragg integer. In equation 5, wave dispersion is effective
only along dB a direction perpendicular to the planes. Freed from the
direct considerations of 3-dimensional lattice periodicity, Bragg’s law
also dispenses of Laue’s desperate conjectures including the infamous
‘missing spot’ proposal, invoked to interpret the patterns recorded
by Friedrich and Knipping [12]. Unsurprisingly Bragg’s formula is
computationally frugal. Once again Alver Gulstrand [11] lauded,

“…It was by a stroke, brilliant in its simplicity, that the Englishman
W.L. Bragg succeeded in replacing von Laue's comparatively complicated
theory of the effect of the crystal lattice by an extremely manageable
formula …”.

In classical crystallography Bragg and Laue’s formulations are
reconciled by imposing a set of stringent selection rules in this
dual space, such that each Bragg reflection spot is associated with a
reciprocal lattice node. In his publications Bragg introduced Miller
indices for calculating the inter-plane separation distance d Bragg.
Reciprocal lattice parameters remain central in crystallography. Legacy
of the dual space is also apparent in textbooks such as Solid State
Physics, (p 99-100) by Neil W Ashcroft, N David Mermin [25] where
it is explained that “a Laue diffraction peak corresponds to a change in
wave vector given by the reciprocal lattice vector K corresponds to a
Bragg reflection from the family of direct lattice planes perpendicular
to K. The order, n, of the Bragg reflection is just the length of K divided
by the length of the shortest reciprocal lattice vector parallel to K.”
Similarly in Principles of the Theory of Solids, the author JM Ziman
[26] writes, (pp: 52-54) that “To satisfy these geometrical conditions in
reciprocal space we construct the Ewald sphere with radius OP equal to
the incident wave-vector …”. It is crucial factor for the convergence of
Laue’s diffraction with Bragg’s reflection is that the scattering system
be perfectly crystalline and of sufficiently large extension to posses a
reciprocal space.

Quasi-Crystals Challenge Bragg

For almost three-quarters of the last century Bragg’s eponymous
formula would remain famously unchallenged. Nevertheless,
particularly because of its heuristic or semi-empirical rationale, many experts take Bragg’s Law as an ansatz to Laue’s diffraction theory; for
instance the noted author Charles Kittle writes “[Bragg] is simple but is
convincing only because it reproduces the results of Laue” [27]. Bragg’s
law was severely pushed back from the experimental side as well by
Dan Shechtman discovery of the illicit five-fold-symmetry in electron
diffraction patterns of some rapidly quenched metallic alloys. Upon his
initial observation, Dan Shechtman, had reportedly blurted out in his
native Hebrew ka,"Eyn chaya zo," (there can be no such creature).

These quasi-crystals are out of bounds of the conventional Bravais
lattice classifications and the spots for these puzzling systems cannot
be Miller or Bravais indexed. Consequently, neither reciprocal lattice
concepts nor Bragg’s law are applicable to quasi-crystals. Eventually the
accumulated experimental evidence compelled the International Union
the International Union of Crystallography to a redefinition of crystals
[13]. However, the occurrence of the forbidden symmetry would be
too perplexing for some of the leading authorities in crystallography,
in particular the two-time Nobel winning chemist Linus Pauling.
“Danny Shechtman is talking nonsense, there are no quasi-crystals, just
quasiscientists."

- Linus Pauling

Lessons from History

The 3-d diffraction grating failed to correctly predict the spots
pattern. But Laue’s bottom-up theory is based on the tiniest periodic
structure, hence the most diffractive part of the whole system– Max
Laue’s scheme zeroes in on the most relevant length scale. In this sense,
Laue was absolutely right- a small organization of atoms and a large
crystal with perfect translational order would both produce spots, there
can be no question about that. Laue missed important details but had
a very fruitful idea. So our first lesson should be as follows-“It is more
important to be fruitful than correct” - A.N. Whitehead On the other
hand Bragg’s heuristic law of two-dimensional atomic planes correctly
solve structure of perfect crystals. Coincidentally, in 1914 Max von
Laue was awarded the Nobel Prize for physics with the official citation,
"for his discovery of the diffraction of X-rays by crystals". Apparently
Laue was fully cognizant of the success of the interference description
and titled his Nobel Lecture [10] as “Concerning the Detection of X-ray
Interferences”! A system with long-range order may not exist without
short-range order, although localized orderly regions may co-exist in
a disordered system. As a result Bragg’s law fails in quasi-crystal, i.e.,
systems with short-range order that lack longrange organization.

This brings us to our second lesson, the principle of ‘reflectiondiffraction
duality’, that is regardless of the mechanism that bring the wavelets or contribute to the deviation from the rectilinear wave
propagation, maxima are obtained only when (where) waves with the
correct phase conditions constructively superpose. Hence, diffraction
explains spots from quasi-crystals whereas specular reflection is the
description for perfect crystals. As a matter of fact, without explicitly
invoking reflection-diffraction duality this idea of is often implied; for
instance the section on ‘crystal diffraction’ a celebrated textbook [28]
describes the relevant physics of wave scattering process as reflection,
without ever using the word ‘diffraction’, not even once in the entire text!

Summary

Upon interaction with atoms in a specimen, waves of appropriate
phase difference give rise to maxima; the geometric pattern of the
maxima provides a faithful map of the spatio-temporal distribution
of the atoms. The famous Bragg’s law of classical crystallography is
extremely facile and productive but requires perfect crystal specimens.
Furthermore lacking a firm theoretical foundation has been a weakness
of Bragg’s law. Structure analysis without Bragg’s law is possible in
principle but correctly taking into account the diffraction of a large
number of atoms requires large computing power. Incorporation of
reflection-diffraction duality may permit one to treat coherent wave
scattering more economically even in the Laue limit. One strategy
is to start with Laue’s three diffraction equations and without the
use of the reciprocal lattice mathematically derive the geometrically
correct solution. Such a solution will be (i) based on a solid theoretical
foundation, (ii) applicable to systems with both short and long
range orders, plus (iii) provide benchmarks for comparisons with
experimental data. Additionally a fresh understanding is likely to
benefit current and future trends toward high intensity radiation, short
exposure, decreased order and specimen size plus emphasis on real
space visualization.